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In this paper we find regular Stein neighborhoods for a union of totally realplanes $M=(A+iI)\mathbb{R}^2$ and $N=\mathbb{R}^2$ in $\mathbb{C}^2$ providedthat the entries of a real $2 \times 2$ matrix $A$ are sufficiently small. Akey step in our proof is a local construction of a suitable function $\rho$near the origin. The sublevel sets of $\rho$ are strongly Levi pseudoconvex andadmit strong deformation retraction to $M\cup N$.

On regular Stein neighborhoods of a union of two totally real planes in $\mathbb{C}^2$